A Core Example with Four Consumers
In this example we’re going to have four consumers, indexed by i ∈ N = {1, 2, 3, 4}. The
two “odd” consumers (i = 1, 3) will be identical and the two “even” consumers (i = 2, 4) will
be identical. The example will show how it is that adding more consumers to the economy
can remove some allocations from the core, in a certain sense, thus shrinking the core to a
smaller set of allocations.
Each consumer in the example has the same utility function, u(x, y) = xy. The two odd
consumers are each endowed with one unit of the y-good and none of the x-good, and each
of the even consumers is endowed with one unit of the x-good and none of the y-good:
(x̊1 , ẙ 1 ) = (x̊3 , ẙ 3 ) = (0, 1)
and
(x̊2 , ẙ 2 ) = (x̊4 , ẙ 4 ) = (1, 0).
The allocations for this economy are in R8+ , so we’re not going to get very far trying to use an
Edgeworth box-type diagram, which will be six-dimensional. But we can exploit the fact that
there are only two distinct “types” of consumer, the odds and the evens, by first analyzing
an economy in which there is only one consumer of each type, say just i = 1 and i = 2. In
this two-person economy it’s easy to see that the Pareto allocations are the ones in which
x1 = y 1 and x2 = y 2 — i.e., the diagonal of the Edgeworth box (EB). The unique Walrasian
equilibrium (WE) has px = py and each consumer receives the bundle (xi , y i ) = ( 21 , 12 ). See
Figure 1.
Now returning to the four-person economy, the Walrasian equilibrium still has px = py , and
each consumer still receives the bundle (xi , y i ) = ( 12 , 21 ). The Pareto allocations are still the
ones in which xi = y i for each consumer. So the Edgeworth box is still somewhat useful
here: in the WE, each consumer of a given type receives the bundle he would receive in the
Edgeworth box economy, and the Pareto set is similar to the Pareto set in the EB economy.
What about the core in the two economies? In the two-person economy, with one consumer
of each type, the core is the entire diagonal of the box. In particular, the lower-left-corner
allocation, in which (x1 , y 1 ) = (0, 0) and (x2 , y 2 ) = (1, 1), is in the core. In the corresponding
four-person allocation, both odd consumers receive (0, 0) and both even consumers receive
(1, 1). We’ll denote this allocation by (b
xi , ybi )N , where N = {1, 2, 3, 4}. See Figure 1. Is
(b
xi , ybi )N in the core of the four-person economy? Let’s figure it out.
(The corner allocation, in which the odd consumers each get (0, 0), may seem kind of extreme.
Before we’re done we’ll see that the argument we’re going to develop applies to plenty of notso-extreme allocations too. In fact, to all but the WE allocation, where each (xi , y i ) = ( 12 , 12 )!)
Figure 1
How can we show that a given allocation, such as (b
xi , ybi )N , is in the core, or alternatively
that it’s not in the core? If we can find just one coalition S that can unilaterally improve on
(b
xi , ybi )N , then it’s not in the core. Conversely, in order to show that the allocation is in the
core, we have to show that none of the fifteen possible coalitions can unilaterally improve on
it.
It’s clear that the four-person “coalition of the whole,” S = N , can’t improve on (b
xi , ybi )N ,
because (b
xi , ybi )N is Pareto efficient. It’s also clear that none of the one-person coalitions
{1}, {2}, {3}, or {4} can improve, because each consumer is receiving at least as much utility
in (b
xi , ybi )N as she receives by just consuming her initial bundle.
For most of the remaining coalitions (i.e., the two- and three-person coalitions), it’s also not
too difficult to show that the coalition can’t improve on (b
xi , ybi )N . For example, the coalition
S = {1, 3} consisting of just the two odd consumers owns two units of the y-good but none
of the x-good, so even though each of these two consumers receives ui = 0 in (b
xi , ybi )N , they
can’t do any better than that with just their own resources.
However, there’s a more systematic way to determine whether coalitions can improve on a
given allocation such as (b
xi , ybi )N : we can use the coalition’s utility frontier. Recall from our
Utility Frontier notes that we used the notation F to denote the set of feasible allocations
P
for the set N of all the consumers — i.e., the allocations that satisfy i∈N xi 5 x̊. We used
P to denote the set of Pareto allocations, and we used U to denote the function that maps
allocations (xi )N to their resulting utility profiles (ui )N = U ((xi )N ) = (u1 (x1 ), . . . , un (xn )).
In the space Rn of utility profiles, U (F) is the set of feasible utility profiles — the image
under U of the set of feasible allocations — and U (P) is the utility frontier, the image under
U of the set of Pareto efficient allocations.
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Let’s replace the set N everywhere in the preceding paragraph with a coalition S ⊆ N , and
P
write FS for the set of allocations (xi )S to S that are feasible for S — i.e., i∈S xi 5 x̊S —
and write PS for the set of allocations to S that are Pareto efficient for S using just its own
resources. Now consider an allocation (b
xi )N to N and the associated utility profile (b
ui )N .
A coalition S can unilaterally improve upon (b
xi )N exactly if there is some allocation (xi )S
to S that’s feasible for S and a Pareto improvement (for S!) upon (b
xi )S — in other words,
S can unilaterally improve if S’s part of the profile (b
ui )N , namely (b
ui )S , is in U (FS ) but
not on the utility frontier, U (PS ). If the utility frontier for S is described by the equation
hS ((ui )S ) = c for some real number c and some strictly increasing function hS , then S can
unilaterally improve upon (b
xi )N if and only if
hS ((b
ui )S ) < c ,
(1)
In short, S can unilaterally improve if and only if (b
ui )S lies strictly inside S’s utility frontier.
Let’s see how this works in our four-consumer example. In the Utility Frontier notes we
saw that if a set S of consumers has a total of x̊S units of the x-good and ẙS units of the
y-good, and if every one of the consumers has the utility function u(x, y) = xy, then the
utility frontier is the set of utility profiles that satisfy the equation
X√
p
ui = x̊S ẙS .
(2)
i∈S
Since the coalition of the whole, S = N , has (x̊, ẙ) = (2, 2), the utility frontier for N is
P
√
ui = 2. The proposed allocation (b
xi , ybi )N , in which each odd consumer receives (0, 0)
i∈N
and each even consumer receives (1, 1), yields the utility profile (b
ui )i∈N = (0, 1, 0, 1), which is
on the utility frontier for N — which we knew it must be, because (b
xi , ybi )N is Pareto efficient.
Let’s look at all the other coalitions:
The one-person coalitions, S = {1}, {2}, {3}, {4}:
UFS is ui = u(x̊i , ẙ i ) = 0. Since u
bi = 0 for i odd and u
bi = 1 for i even, u
bi is either on
or outside the utility frontier for each i. So none of these coalitions can unilaterally improve
upon (b
xi , ybi )N .
The two-person coalitions S = {1, 2}, {1, 4}, {3, 2}, {3, 4}:
(See Figure 2.)
Each of these is just like the two-person Edgeworth box economy, with one odd consumer
√
√
and one even, and UFS is given by uodd + ueven = 1. Since u
bodd = 0 and u
beven = 1, we
have (b
uodd , u
beven ) = (0, 1), which is on the utility frontier for S. So none of these coalitions
can unilaterally improve upon (b
xi , ybi )N .
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Figure 2
Figure 3
The two-person coalitions S = {1, 3} and S = {2, 4}:
(See Figure 3.)
In each of these two cases the coalition has none of one of the two goods, so its utility
P √
frontier is given by i∈S ui = 0, and the utility frontier is therefore the singleton {(0, 0)}.
For S = {1, 3}, the utility profile (b
u1 , u
b3 ) is (0, 0), which is on the utility frontier. For
S = {2, 4}, the utility profile (b
u2 , u
b4 ) is (1, 1), which is outside the utility frontier. So neither
of these coalitions can unilaterally improve upon (b
xi , ybi )N .
The three-person coalitions S = {1, 2, 4} and S = {3, 2, 4}:
(See Figure 4.)
Each of these coalitions has one odd consumer and two even consumers; we’ll work with
S = {1, 2, 4}. We have (x̊S , ẙS ) = (2, 1), so the utility frontier for S is given by the equation
√
√
√
√
u1 + u2 + u4 = 2. The utility profile (b
u1 , u
b2 , u
b4 ) = (0, 1, 1) is outside this utility
frontier. The same is true for S = {3, 2, 4}, so neither of these coalitions can unilaterally
improve upon (b
xi , ybi )N .
Figure 4
Figure 5
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The three-person coalitions S = {1, 2, 3} and S = {1, 3, 4}:
(See Figure 5.)
Each of these coalitions has one even consumer and two odd consumers; we’ll work with
S = {1, 2, 3}. We have (x̊S , ẙS ) = (1, 2), so the utility frontier for S is given by the equation
√
√
√
√
u1 + u2 + u3 = 2. The utility profile (b
u1 , u
b2 , u
b3 ) = (0, 1, 0) is inside this utility frontier.
The same is true for the coalition S = {1, 3, 4}, so either of these coalitions can unilaterally
improve upon (b
xi , ybi )N .
To summarize, we’ve found two coalitions that can unilaterally improve upon the proposed
allocation (b
xi , ybi )N , and therefore (b
xi , ybi )N is not in the core of the four-person economy.
While we haven’t actually found a feasible allocation (xi , y i )S for a coalition S that’s an
improvement on (b
xi , ybi )N for S, we nevertheless know that one exists, and that’s enough to
tell us that (b
xi , ybi )N is not in the core.
Here’s an allocation (e
xi , yei )S that’s feasible for S = {1, 2, 3} and that makes each consumer
in S strictly better off than at (b
xi , ybi )N :
( (e
x1 , ye1 ), (e
x2 , ye2 ), (e
x3 , ye3 ) ) = ( 18 , 14 ), ( 43 , 32 ), ( 81 , 14 ) .
1 9 1
, 8 , 32 ), which is larger in each
This allocation yields the utility profile (e
u1 , u
e2 , u
e3 ) = ( 32
component than (b
u1 , u
b2 , u
b3 ) = (0, 1, 0).
Notice that there are a lot of allocations to N near (b
xi , ybi )N that are also unilaterally improved
upon by S = {1, 2, 3} with the S-allocation ( (e
x1 , ye1 ), (e
x2 , ye2 ), (e
x3 , ye3 ) ) = ( 81 , 14 ), ( 43 , 23 ), ( 18 , 41 ) .
For example, consider the allocation
( (x1 , y 1 ), (x2 , y 2 ), (x3 , y 3 ), (x4 , y 4 ) ) = ( 16 , 61 ), ( 56 , 65 ), ( 61 , 16 ), ( 65 , 65 ) .
The S-allocation (e
xi , yei )S makes each consumer strictly better off. So any allocation to N
that gives each of the odd consumers
1
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unit of each good, or less, is not in the core.
Figure 6
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Exercise 1: Exactly which allocations of the form
( (x1 , y 1 ), (x2 , y 2 ), (x3 , y 3 ), (x4 , y 4 ) ) = ( (a, a), (b, b), (a, a), (b, b) )
are in the core of the four-consumer economy?
Exercise 2: Now assume there are r consumers of each type, with the utility function and
initial bundles in the example. Determine which allocations of the form in Exercise 1 are
core allocations. This requires two steps. The algebra in Step 1 is more difficult; but with
Step 1 in hand, Step 2 is easy.
Step 1: Represent each coalition S by the pair (r, k), where r is the number of odd members
and k is the number of even members. You need to show that the coalitions that can improve
upon the most allocations are the ones in which k = r − 1.
Step 2: Determine for each r what is the smallest value of a for which the allocation in
Exercise 1 can’t be improved upon by a coalition with composition (r, r − 1).
Notice how quickly the core becomes very small, if the only core allocations are the ones that
have the form in Exercise 1. (And these are indeed the only core allocations, as we’ll see
shortly.)
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